Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 11 / ii / Solution

Expanding the squared Euclidean norm and applying the Cauchy-Schwarz inequality givesBoth sides are nonnegative, so taking square roots proves the triangle inequality, $\boxed{|a+b|\le |a|+|b|}$. For nonzero vectors, equality requires a positive real proportionality factor.